Quantum optics of plasmon polaritons and velocity sum rules

We demonstrate that the reciprocal-space Hamiltonian method for quantizing the electromagnetic fields in a dispersive dielectric medium leads to inconsistencies when applied to the cases of plasmon polaritons and coupled phonon-plasmon polaritons. In particular, we show that the use of the standard expressions for the dielectric functions in these two cases leads to the violation of the Huttner-Barnett velocity-ratio sum rule and, hence, to the breakdown of the equal-time canonical field commutation relations. We show that correct behavior is recoverable after the dielectric functions are appropriately regularized. Regularization effectively leads in each case to the introduction of an additional transverse low-frequency polariton branch. This is sufficient to rectify the theory and there is no need to invoke the presence of the longitudinal modes. The theory is illustrated with reference to the coupled phonon-plasmon polaritons of Si-doped GaAs.